The field of the present invention relates to electromagnetic acoustic transducers (EMAT), more particularly, to rotating EMAT""S used as strain gages.
In non-destructive testing using EMAT""s, it is preferable for the stressed region to be struck by the incident wave at an angle of 90 degrees. This requires the EMAT to be oriented, data collected, then re-oriented for the next series of measurements. This practice involves large amounts of time in manually orienting and re-orienting the EMAT. The prior art does not allow for the rapid collection and analysis of data for calculation of stress in a specimen.
Further, the prior art involved measurement of signals on a specimen at a particular point in time. The measurements reflected certain characteristics of the specimen at the time the measurements were taken. Characteristics detected primarily included defects and flaws. Real time detection of the stress state of a specimen was not available.
Representative of the art is:
U.S. Pat. No. 5,813,280 (1998) to Johnson et al. discloses a force sensor including a cylindrical body having a central section and two distal sections wherein selected acoustic resonant modes are trapped in the central section and decays exponentially in the distal sections possibly using an electromechanical acoustic transducer (EMAT) to excite and detect the selected resonant modes in the central section.
U.S. Pat. No. 5,808,201 (1998) to Hugentobler discloses an improved electromagnetic acoustic transducer (EMAT) for monitoring uniaxially applied stress in an underlying workpiece.
U.S. Pat. No. 5,750,900 (1998) to Hugentobler et al. discloses an improved strain gauge and a method of using an electromagnetic acoustic transducer (EMAT) for monitoring stress and strain in an underlying workpiece.
U.S. Pat. No. 5,652,389 (1997) to Schapes et al. discloses a method and apparatus for the non-contact inspection of workpieces having a plate-like portion of the first part joined via an inertia weld to the end of a second part extending away from the plate-like portion.
U.S. Pat. No. 5,299,458 (1994) to Clark, Jr. et al. discloses an indication of the formability of metallic sheet which is determined using a correlation between nondestructively measurable ultrasonic properties and a formability index.
U.S. Pat. No. 5,115,681 (1992) to Bouheraoua et al. discloses a two-step method of studying the acoustic response of a piece by recording signals with acoustic sensors.
U.S. Pat. No. 5,811,682 (1998) to Ohtani et al. discloses an electromagnetic acoustic transducer for magnets and a sheet type coil unit.
U.S. Pat. No. 5,804,727 (1998) to Lu et al. discloses a method for determining and evaluating physical characteristics of a material.
U.S. Pat. No. 5,714,688 (1998) to Buttram et al. discloses a method of examining a ductile iron casting to determine a percent of nodularity present in the casting using an electromagnetic acoustic transducer (EMAT) system to determine a time-of-flight of an ultrasonic shear wave pulse transmitted through the casting at a selected location from which a velocity of sound in the casting can be determined.
U.S. Pat. No. 5,675,087 (1997) to MacLauchlan et al. discloses a device for measuring a load on a part and for monitoring the integrity of the part such as a bolt, comprises a socket having walls defining an interior space wherein the socket engages the bolt for transmitting a load to the bolt.
U.S. Pat. No. 5,619,423 (1997) to Scrantz discloses an improved system, method, and apparatus for the external ultrasonic inspection of fluidized tubulars and tanks.
U.S. Pat. No. 5,608,691 (1997) to MacLauchlan et al. discloses a shield for an electromagnetic acoustic transducer (EMAT) has multiple layers of electrically insulating and electrically conductive materials which contain a coil of the EMAT.
U.S. Pat. No. 5,537,876 (1996) to Davidson et al. discloses an apparatus and method for nondestructive evaluation for detection of flaws in butt welds in steel sheets using horizontal shear ultrasonic waves generated on the surface thereof.
U.S. Pat. No. 5,467,655 (1995) to Hyoguchi et al. discloses a method and apparatus for measuring properties of a cold-rolled thin steel sheet, comprising an electromagnetic ultrasonic wave device, a computing device, and a controlling device for executing the measurement of properties and the computation.
U.S. Pat. No. 5,237,874 (1993) to Latimer et al. discloses a method and apparatus of non-destructive testing wherein a generally bi-directional wave-generating electromagnetic acoustic transducer is pivotally mounted upon a base with this transducer being continuously rotated or oscillated upon the base as it is moved with respect to the workpiece (or the workpiece is moved with respect to the base).
U.S. Pat. No. 5,164,921 (1992) to Graff et al. discloses an electrodynamic permanent magnet transducer for the non-destructive testing of workpieces by means of ultrasound.
U.S. Pat. No. 5,050,703 (1991) to Graff et al. discloses an electrodynamic transducer head for a non-destructive testing of workpieces with electrically conductive surfaces by way of ultrasound.
U.S. Pat. No. 4,522,071 (1985) to Thompson discloses a method and apparatus for determining stress in a material independent of micro-structural variation and anisotropy""s.
U.S. Pat. No. 4,466,287 (1984) to Repplinger et al. discloses a method for non-contact, non-destructive testing of a test body of ferromagnetic and/or electrically-conductive material with ultrasound waves.
U.S. Pat. No. 4,295,214 (1981) to Thompson discloses an electromagnetic acoustic transducer, including an electrical conductor adapted to carry an alternating current in a current plane.
What is needed is a rotating EMAT, connected to a processor, that can be rotated through 360 degrees. What is needed is a rotating EMAT connected to a processor using an algorithm to calculate birefringence from the normalized differences in the phase of SH waves in a specimen. What is needed is a rotating EMAT connected to a processor using an algorithm to calculate the pure-mode polarization directions of SH waves in a specimen. What is needed is a rotating EMAT connected to a processor using an algorithm to calculate the stress in a specimen. What is needed is a rotating EMAT connected to a processor used as a strain gage.
The primary aspect of the present invention is to provide a rotating EMAT, connected to a processor, that can be rotated through 360 degrees.
Another aspect of the present invention is to provide a rotating EMAT connected to a processor using an algorithm to calculate birefringence from the normalized differences in the phase of SH waves in a specimen.
Another aspect of the present invention is to provide a rotating EMAT connected to a processor using an algorithm to calculate the pure-mode polarization directions of SH waves in a specimen.
Another aspect of the present invention is to provide a rotating EMAT connected to a processor using an algorithm to calculate the stress in a specimen.
Another aspect of the present invention is to provide a rotating EMAT connected to a processor used as a strain gage.
Other aspects of this invention will appear from the following description and appended claims, reference being made to the accompanying drawings forming a part of this specification wherein like reference characters designate corresponding parts in the several views.
The invention comprises an EMAT connected to a processor. The EMAT rotates about a central axis while collecting data on a specimen. The invention is used to measure the change in plane stress in metallic components (e.g. rolled plates of steel and aluminum). In the absence of stress, the pure-mode polarization directions of SH-waves in these components are the rolling and transverse directions. The velocity of a wave polarized in the rolling directions generally exceeds that of a wave polarized in the transverse direction due to anisotropy induced by rolling (which causes preferential alignment of the grains). With the transducer polarized at an arbitrary angle the wave xe2x80x9csplitsxe2x80x9d into components polarized along the rolling and transverse directions. Each component propagates with its own velocity. These components recombine along the transducer polarization direction and interfere. Consider now the case of applied plane stress, having components "sgr"xx, "sgr"yy and "sgr"xy. Because of the stresses there will be two effects, first, the orientation of pure-mode polarization directions may change, second, the velocity or time of flight (TOF) of waves polarized along these directions will change.
In the absence of stress the rolling and transverse directions typically are pure-mode polarization directions for SH-waves. It is convenient to refer stresses to a coordinate system parallel to these axes: see FIG. 2. In general the principal stress directions are not coincident with these axes, and a rotation of the pure mode directions through angle "PHgr" results: see FIG. 5.
The relation between "PHgr" and stress is given by:
tan 2"PHgr"={2m "PHgr"xy}/{Bo+m ("sgr"yyxe2x88x92"sgr"xx)}xe2x80x83xe2x80x83(1)
where m is the acoustoelastic constant and Bo is the birefringence in the unstressed state. The birefringence is defined by
B=(Vfxe2x88x92Vs)/Vavexe2x80x83xe2x80x83(2)
where Vf is the velocity of the faster wave, Vs is the slower wave velocity and Vave is their average. The birefringence is related to stress by:
B2=[Bo+m ("sgr"yyxe2x88x92"sgr"xx)]2+[2m"sgr"xy]2xe2x80x83xe2x80x83(3)
Eqns. (1) and (3) can be combined (using trigonometric identities) to give:
B cos 2"PHgr"xe2x88x92Bo=m ("sgr"yyxe2x88x92"sgr"xx)xe2x80x83xe2x80x83(4)
and
B sin 2"PHgr"=2m"sgr"xyxe2x80x83xe2x80x83(5)
Most applications of the acoustic birefringence method to date have used eqn. (4). That is measurements are made where the principal stress direction is coincident with the rolling and transverse directions, so that (see eqn. (1)), either "PHgr"=0 or "PHgr"=xcfx80/2. Problems arise when the texture is not homogenous so that variations in Bo occur. These variations result in artifacts in stress measurement and if large enough, can give rise to errors larger than the stresses to be determined.
In contrast eqn. (5) shows that measurement of the shear stress ok is independent of texture. By determining gradients of shear stress and using the stress-equilibrium equations it is possible in principle to determine the normal stresses "sgr"yy and "sgr"xx.
To determine stress we require that correct values of B and "PHgr" be obtained. Therefore we seek to determine model with the minimum number of parameters necessary to obtain accurate stress measurements from our ultrasonic data.
This model is implemented in interactive software: to query the operator for inputs; produce graphs to compare data and the fit based on our model. In its current form this algorithm runs on a personal computer with a processor clock rate of 100 MHz. Running the algorithm requires less than 30 s from the time phase data is imported, until the values of B and "PHgr" are calculated.
Amplitude and Phase for Arbitrary Orientation of EMAT Consider the case shown in FIG. 5, where the EMAT is oriented at an angle xcex8 to the X and Y-axes. (These are the pure-mode polarization directions in the stressed state.) These axes are rotated by angle "PHgr" from the pure-mode polarization directions in the unstressed state (Xo-Yo axes). Hence the EMAT is oriented at angle xcex7 to the Xo-axis; xcex7=xcex8+"PHgr".
The EMAT detects the particle velocity. The expression for the particle velocity un in the nth echo is:
un=A*exp{j(xcfx89txe2x88x92Ps)}+B* exp{j(xcfx89t xe2x88x92Pf)}xe2x80x83xe2x80x83(6)
Here Ps is the phase of the component polarized along the X-axis (xe2x80x9cslowxe2x80x9d direction) and Pf is the corresponding phase of the component polarized along the Y-axis (xe2x80x9cfastxe2x80x9d direction). xcfx89 is the frequency (in radians) of the toneburst used to excite the EMAT. Denote d to be twice the specimen thickness. For an acoustic pathlength of nd, the phase is Ps=xcfx89nd/Vs for the slow wave component.
The coefficients A* and B* give the contributions of the components polarized along the X- and Y- axes, respectively:
A*=U exp (xe2x88x92xcex1snd) cos2xcex8xe2x80x83xe2x80x83(7)
B*=U exp (xe2x88x92xcex1fnd) sin2xcex8xe2x80x83xe2x80x83(8)
Here U is the amplitude of the particle velocity generated by the EMAT at the specimen surface; xcex1s and xcex1f are the attenuation coefficients for the xe2x80x9cslowxe2x80x9d and xe2x80x9cfastxe2x80x9d waves.
For simplicity we suppress the exp (jxcfx89t) dependence. The expression for the particle velocity can be written as a phasor with amplitude an
an(xcex8; xcex94Pn)={A*2+B*2+2A*B* cos(Psxe2x88x92Pf)}xc2xdxe2x80x83xe2x80x83(9)
and phase Pn
Pn=tanxe2x88x921{(A*sin Ps+B*sin Pf)/(A*cos Ps+B*cos Pf) }xe2x80x83xe2x80x83(10)
Now consider the special case xcex1s=xcex1f. That is, we assume the attenuation is the same for components polarized in the slow and fast directions. We rewrite the expressions for amplitude and phase in forms to show more clearly the effect of interference between these components. We denote xcex94Pn=(Psxe2x88x92Pf) as the difference in phase between the slow and fast wave components in the nth echo.
an(xcex8; xcex94Pn)=U exp (xe2x88x92xcex1nd){1xe2x88x92sin2xcex8sin2(xcex94Pn/2) }xc2xdxe2x80x83xe2x80x83(11)
The phase is now given by Pn=(Pa,n+"psgr"n). Here Pa,n is the average phase in the echo (the phase for no birefringence), and "psgr"n is the additional phase due to interference between fast and slow wave components:
"psgr"n(xcex8; xcex94Pn)=tanxe2x88x921{tan (xcex94Pn/2) cos 2xcex8}.xe2x80x83xe2x80x83(12)
We used a commercial instrument which recorded both amplitude and phase data as our EMAT rotated. The instrument, under computer control, produced a file of relative amplitude data as a function of xe2x80x9cclock timexe2x80x9d, t. Data collection was started so that t=0 when xcex7=0; that is, with the EMAT oriented along the transverse direction (see FIG. 2). The EMAT is rotated with constant angular velocity xcexa9. Since xcex7=xcexa9t, we generated a data file with relative amplitude and phase as dependent variables and EMAT orientation xcex7 as the independent variable.
The relative amplitude AR is given by:
AR (xcex8; xcex94Pn)=20 log10{a (xcex8; xcex94Pn)/ a (xcex8; xcex94Pn)}xe2x80x83xe2x80x83(13)
FIG. 20 shows the theoretical behavior of AR for different values of xcex94Pn. Note that maxima occur for xcex8=0xc2x0, 90xc2x0, etc.; that is, with the EMAT oriented along the slow and fast directions. The minima occur at xcex8=45xc2x0,135xc2x0, etc.; now the EMAT is oriented midway between slow and fast directions. The minima become more pronounced as xcex94Pn approaches 180xc2x0.
The behavior of the additional phase term "psgr"n (xcex8; xcex94Pn) is shown in FIG. 21. For small differences in phase between fast and slow wave components, "psgr"n behaves sinusoidally. As the difference approaches 180xc2x0, the phase term behaves like a square wave.
To determine "PHgr", the algorithm first calculates xe2x80x9csynthetic dataxe2x80x9d "psgr"nsyn
"psgr"nsyn (xcex7xe2x88x92"PHgr"*; xcex94Pn)=tanxe2x88x921{tan(0.5[Psxe2x88x92Pf]cos 2(xcex7xe2x88x92"PHgr"*)}xe2x80x83xe2x80x83(14)
where "PHgr"* is an assumed value for "PHgr" (recall that xcex8=xcex7xe2x88x92"PHgr"). The algorithm next calculates the difference between the data, and values predicted from eqn. (14), for each value of xcex7. The phase prediction errors are squared and summed to give the sum of squared residuals, for the assumed value of "PHgr".
The estimate of "PHgr" is constrained to fall in the interval (0,xcfx80). The best estimate of "PHgr" is determined by minimizing the sum of the squared phase residuals.
This method gives good results for measurements made on an aluminum shrink-fit specimen with a large EMAT. In this case both the relative amplitude and phase data behaved as predicted by eqn. (11) and (13). However, for the small EMAT, discrepancies occurred such as those shown in FIGS. 22 and 23.
For the relative amplitude data the local maxima are not of equal magnitude. This indicates that the attenuation is not the same for waves polarized along the fast and slow directions. The local minima are also of unequal magnitude. This implies that the minima do not repeat when the EMAT rotates by 180xc2x0.
Likewise the phase data does not repeat under 180xc2x0 rotation of the EMAT. This would imply that rotating the EMAT by 180xc2x0 somehow changes the velocity, which is nonphysical. In addition the rate of change of phase with rotation appears to have asymmetry, another result not in agreement with the 3-parameter model predictions.
To improve the agreement between model predictions and experiment, we assumed that the attenuation in the fast and slow waves is different. Differential attenuation can be quantified by the parameter xcex94An: xcex94An=[exp(xe2x88x92xcex1fnd)xe2x88x92exp (xe2x88x92xcex1snd)]/2. If xcex94An greater than 0, the attenuation in the slow wave is larger; if xcex94An less than 0 the attenuation in the fast wave is larger.
This model predicts that the local maxima of the relative amplitude still occur for the fast and slow directions, when xcex94Pn is approximately equal to an odd multiple of xcfx80. However if xcex94An greater than 0, AR will be larger for the fast direction and vice versa. This is in qualitative agreement with the data of FIG. 22.
The model also predicts that the local minima will be shifted from xcex8=xcfx80/4,3xcfx80/4, etc. to xcex8=xcfx80/4xe2x88x92xcex94An/2,3xcfx80/4+xcex94An/2, etc.[8]. Hence if xcex94An greater than 0, the model predicts that maximum interference between the fast and slow wave components no longer occurs with the EMAT oriented midway between the fast and slow directions. Since the slow wave component is smaller, the maximum interference occurs with the EMAT oriented closer to the slow direction by xcex94An/2 radians.
The 4-parameter model predicts shifts in the zeroes of the additional phase "psgr"n from xcex8=xcfx80/4, 3xcfx80/4,etc. (halfway between slow and fast directions) to xcex8=xcfx80/4xe2x88x92xcex94An/2, 3xcfx80/4+xcex94An/2,etc. The locations of maxima and minima in "psgr"n remain the same as in the 3-parameter fit.
A nonlinear least squares fitting routine was used to implement the 4-parameter model. Since the 4-parameter model can account for the difference in local maxima of AR it improved the agreement with relative amplitude data. Since it cannot account for differences in measured "psgr"n under 180xc2x0 rotation it did not significantly improve the fit to phase data.
In this case it is assumed that the center of the EMAT aperture is not coincident with the center of rotation. A simple model of EMAT transduction is used to show that the EMAT output voltage is proportional to the particle velocity at the center of the aperture. An instrument is used which measures phase and amplitude of the output voltage. Hence the aperture center.
Assume that the center of the aperture is offset an amount ro from the center of rotation, and at an angle xcex1 to the EMAT polarization direction. Consider a coordinate system with origin at the center of rotation and axes parallel to the Xo- and Yo- directions. In this coordinate system the coordinates of the aperture center are (ro cos(xcex7+xcex1), rosin (xcex7+xcex1)).
Assume that the velocities of the slow and fast waves are inhomogeneous. Then for example the slow velocity Vs can be expanded in a (truncated) Taylor series:
Vs (ro cos xcex7, ro sin xcex7)=Vs (0,0)+(∂Vs/∂Xo) ro cos (xcex7+xcex1) +(∂Vs/∂Yo) ro sin (xcex7+xcex1).xe2x80x83xe2x80x83(15)
Since Ps=xcfx89nd/Vs, we model the slow phase as follows:
Ps,m=xcex32+xcex35 cos xcex7+xcex36 sinxcex7xe2x80x83xe2x80x83(16)
Here Ps,m is the measured phase and xcex32 is the correct phase (phase at the center of rotation). A similar expression can be written for Pf:
xe2x80x83Pf,m=xcex33+xcex37 cos xcex7+xcex38 sin xcex7xe2x80x83xe2x80x83(17)
xcex94Pn is given by xcex32xe2x88x92xcex33; that is, xcex94Pn is the value of Ps,mxe2x88x92Pf,m averaged over 0xe2x89xa6xcex7xe2x89xa62xcfx80.
In like fashion we account for inhomogeneity in differential attenuation by rewriting eqn. (7) as
A*=(xcex34+xcex39cos xcex7+xcex3100 sin xcex7) cos2 (xcex7xe2x88x92xcex31),xe2x80x83xe2x80x83(18)
where we use xcex8=xcex7xe2x88x92"PHgr", and let "PHgr"=xcex31. For example, xcex34 is the value of exp(xe2x88x92(xcex1sxe2x88x92xcex1f)nd) at the center of rotation. The same procedure which leads to eqn. (18) also gives B*=sin2(xcex7xe2x88x92xcex31).
From the above we see that this model requires a total of 10 parameters (6 for phase; 3 for attenuation; 1 for "PHgr").
Improved agreement was found between model predictions and the phase data; see FIG. 24. The agreement between predicted and measured relative amplitude also improved, but to a lesser degree.
In this model we kept the same functional forms for A*, B*, and the phase "psgr"n. How ever we now assumed that the amplitude is given by
a(xcex8;xcex94Pn)={f(xcex7) [A*2+B*2+2A*B* cos(Psxe2x88x92Pf)]}xc2xdxe2x80x83xe2x80x83(19)
where
f(xcex7)=xcex311+xcex312 cos xcex7+xcex313 sin xcex7xe2x80x83xe2x80x83(20)
The term f(xcex7) allows for variation in apparent transducer gain as the EMAT rotates.
The predicted phase, based on 13-parameter model, is now almost indistinguishable from the phase data. There was also improvement in the fit to the amplitude data using this model; see FIG. 25.
The relation between the stress changes and changes in (weighted) time-of-flight (TOF""s) is given by:
xcex94"sgr"xx=K11xcex94xcfx841+K12xcex94xcfx842xe2x80x83xe2x80x83(21)
xcex94"sgr"yy=K21xcex94xcfx841+K22xcex94xcfx842
xcex94"sgr"xy=K33xcex94xcfx843
Here we define:
xcfx841=(Ts cos2"PHgr"+Tf sin2"PHgr")/Toxe2x80x83xe2x80x83(22)
xcfx842=(Tf cos2"PHgr"+Ts sin2"PHgr")/To
xcfx843=[(Tsxe2x88x92Tf) sin22"PHgr"]/2To.
"PHgr" is the angle between the pure-mode directions (xe2x80x9cacoustic axesxe2x80x9d) in the unstressed and stressed states (see FIG. 2). Ts is the TOF of the wave polarized along the xe2x80x9cslowxe2x80x9d direction (X-axis) and Tf is the corresponding TOF in the xe2x80x9cfastxe2x80x9d direction (Y-axis); To is the average TOF. The K""s are acoustoelastic constants determined by performing uniaxial tension tests with specimens cut at a different angle to the specimen rolling direction. For example:
K11=(dxcfx841/d"sgr"xx)xe2x88x921,
K12=(dxcfx842/d"sgr"xx)xe2x88x921
for a uniaxial specimen cut along the transverse direction. Once the K""s are known (either from values in the literature or by performing the uniaxial tension tests described above), the procedure to determine stress changes proceeds as follows. The transducer is placed at selected measurement locations on the specimen in the initial state. TOFs and "PHgr" are then measured (note: if the initial state is unstressed, then "PHgr" is zero and it is only necessary to measure TOFs). The specimen is then stressed; at some convenient time later the EMAT is repositioned at the same locations and a new set of TOFs and "PHgr" measured. The EMAT is used to generate and receive waves that are used to determine "PHgr", in conjunction with either a phase-sensitive instrument or time-measuring instrument to measure TOF""s. The EMAT is also used to generate and receive waves polarized along the pure-mode directions to determine the TOFs.
Values of xcfx84""s are calculated for both stressed and unstressed state, and stress changes calculated. Since the K""s are large, small errors in TOFs result in large stress errors. Conversely, stress causes only small changes in TOFs. An error analysis shows that to resolve 10 MPa of stress in steel requires resolution in TOF of order of 10 ppm.
The transduction mechanism of the EMAT makes it ideal for this application. Conventional piezoelectric transducers require couplants to transmit sound from transducer to specimen. It is difficult to accurately control the couplant thickness, and special fixturing is required. In contrast EMATs generate sound directly in the surface of the specimen and can work with clearances of a few mm. Hence they can be rapidly scanned and rotated to collect the required data.
A cross-section of the EMAT shows that it consists of a xe2x80x9cracetrackxe2x80x9d coil, excited by a high-power toneburst by the commercial instrument. Magnets (with opposite polarity) are placed above the xe2x80x9cstraightxe2x80x9d sides of the coil. The reaction force FL between the magnetic induction B with the induced eddy current density J is: FL=Jxc3x97B. This causes a shearing force at the specimen surface, which sets up a propagating SH wave.
Implementation of the motorized, rotating EMAT is as follows. The coil and magnet are contained inside a rotating cylinder. Bearings are press-fit onto the cylinder and are seated inside the (fixed) outer case. The motor is mounted on the case and rotates the cylinder by means of a gear train. Offsetting the motor in this manner allows rotation of the EMAT without twisting the signal and power cables which connect the EMAT to the commercial instrument.
The specifications for the EMAT are: effective EMAT aperture is 25xc3x9725 mm; angular velocity =1 rpm.
As an example, the EMAT generates a wave polarized at angle "THgr" to the pure-mode polarization directions in the stressed configuration. The EMAT is polarized at a known angle xcex7 to the Xo-axis (the transverse direction). The wave splits into components polarized along the X,Y axes, having amplitude cos xcex8, sin xcex8, respectively. Each component of the wave propagates with a different velocity. Upon recombining the components along the transducer polarization direction, the particle velocity U is given by:
U2xcx9cexp (j Pave){cos2xcex8exp (j xcex94P)+sin2xcex8exp (xe2x88x92j xcex94P)}xe2x80x83xe2x80x83(23)
Here Pave is the average phase and xcex94P=Psxe2x88x92Pf is difference in phase between velocity components polarized along slow and fast directions. The term in braces represents the effect of interference between these components. It is this term that is used to determined.
After some algebraic manipulation, this term is re-written in the form a(xcex8) exp(j"PHgr"(xcex8)) where "PHgr"(xcex8) is the additional phase resulting from interference. "PHgr"(xcex8) is determined from measurements made with the phase-sensitive commercial instrument as the EMAT is rotated. We calculated
"PHgr"(xcex8)=arctan {tanxe2x88x921 (xcex94P/2) cos 2xcex8}
Note that xcex8 is unknown, since xcex8=xcex7xe2x88x92"PHgr" and "PHgr" is unknown.
"psgr"(xcex7) is then measured with the phase sensitive instrument as the EMAT is rotated. Software supplied with the instrument generates a file containing xe2x80x9cclock timexe2x80x9d and phase. Clock time is recorded as time since the EMAT is polarized along the transverse axis. The file is recorded on the computer hard disk for further evaluation by our algorithm.
The operator then imports the data and runs the algorithm. In the first step, clock time is converted to angle xcex7 by: xcex7=xcfx89t where xcfx89 is the angular velocity of the rotating EMAT. The algorithm next determines the maximum and minimum phase "psgr"max and "psgr"min. The difference in phase xcex94P is calculated from: xcex94P="psgr"maxxe2x88x92"psgr"min 
From values of xcex7 and xcex94P the xe2x80x9csynthetic phasexe2x80x9d is calculated:
"psgr"syn (xcex7;"PHgr"*)=arctan {tanxe2x88x921(xcex94P/2) cos 2(xcex7xe2x88x92"PHgr"*)}
Note that xcex8=xcex7xe2x88x92"PHgr";here "PHgr"* is the assumed value of "PHgr". For each value of "PHgr"* (in 1xc2x0 increments) in the interval between 0 and 180 degrees, the algorithm calculates the residual:
R("PHgr"*)=xcexa3("psgr"syn (xcex7i;"PHgr"*) xe2x88x92"psgr"(xcex7i;"PHgr"*) 2}.
Here xcex7I=xcfx89tI where tI is the Ith increment of clock time. The algorithm searches through values of R("PHgr"*)to find the minimum. The corresponding value of "PHgr"* gives the best estimate of "PHgr".
The EMAT is rotated and phase data is recorded for about 80 seconds with the 1 rpm motor. The algorithm requires about 30 seconds to run from the time the phase data is imported until the correct "PHgr" is determined.
Once "PHgr" is determined, the EMAT is rotated so that it is polarized along, for example, the X-axis. We then measure TOF of the wave polarized along this axis with our digital gate/counter system. The operator then selects a particular cycle on a particular echo for measurement purposes. These must be the same for measurements taken in the unstressed and stressed states. This is done by manually adjusting the location (in time) of the gate such that it is in the ON state just before the zero crossing. Then a comparator circuit generates a square pulses when the zero crossing occurs. This pulse is input to the STOP channel of the counter. The counter is started by a trigger pulse from the commercial instrument. The trigger pulse is coincident with the toneburst that drives the EMAT.
The EMAT is positioned at the measurement location and is rotated; the phase data is measured and recorded by the commercial instrument; the angle "PHgr" is determined with the algorithm. Then the EMAT is rotated along the xe2x80x9cfastxe2x80x9d and xe2x80x9cslowxe2x80x9d directions and the corresponding TOFs measured with the counter.